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Review radius for satellite in geostationary orbit

step inference rule input feed output step validity (as per SymPy)
1
  • 0000111777: change four variables in expression
  • number of inputs: 1; feeds: 8; outputs: 1
  • Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ in Eq.~\ref{eq:#9}; yields Eq.~\ref{eq:#10}.
  1. 6935745841
    \(F = G \frac{m_1 m_2}{x^2}\)
  1. 7819443873
    \(r\)
  1. 4830480629
    \(x\)
  1. 3486213448
    \(m_{\rm satellite}\)
  1. 3088463019
    \(m_2\)
  1. 4153613253
    \(m_{\rm Earth}\)
  1. 9794128647
    \(m_1\)
  1. 3594626260
    \(F_{\rm gravity}\)
  1. 3398368564
    \(F\)
  1. 5563580265
    \(F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
 LHS diff is -pdg0002867 + pdg0004202 RHS diff is pdg0004851*pdg0005022*pdg0006277/pdg0004037**2 - pdg0003569*pdg0005458*pdg0006277/pdg0002530**2
2
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 9226945488
    \(F = \frac{m v^2}{r}\)
 no validation is available for declarations
3
  • 0000111236: change three variables in expression
  • number of inputs: 1; feeds: 6; outputs: 1
  • Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\ref{eq:#7}; yields Eq.~\ref{eq:#8}.
  1. 9226945488
    \(F = \frac{m v^2}{r}\)
  1. 9789485295
    \(v_{\rm satellite}\)
  1. 7912578203
    \(v\)
  1. 2114570475
    \(m_{\rm satellite}\)
  1. 3342155559
    \(m\)
  1. 1333474099
    \(F_{\rm centripetal}\)
  1. 5089196493
    \(F\)
  1. 4627284246
    \(F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}\)
 LHS diff is -pdg0001687 + pdg0004202 RHS diff is (pdg0001357**2*pdg0005156 - pdg0003569*pdg0004082**2)/pdg0002530
4
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 3176662571
    \(F_{\rm centripetal} = F_{\rm gravity}\)
 no validation is available for declarations
5
  • 0000111732: substitute LHS of two expressions into expression
  • number of inputs: 3; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 3176662571
    \(F_{\rm centripetal} = F_{\rm gravity}\)
  1. 4627284246
    \(F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}\)
  1. 5563580265
    \(F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
  1. 4072200527
    \(\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
 recognized infrule but not yet supported
6
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 6785303857
    \(C = 2 \pi r\)
  1. 3236313290
    \(d\)
  1. 1823570358
    \(C\)
  1. 9262596735
    \(d = 2 \pi r\)
 LHS diff is -pdg0001943 + pdg0003034 RHS diff is 0
7
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 5426308937
    \(v = \frac{d}{t}\)
  1. 9262596735
    \(d = 2 \pi r\)
  1. 4245712581
    \(v = \frac{2 \pi r}{t}\)
 LHS diff is -pdg0001357 + pdg0001943 RHS diff is 2*pdg0002530*pdg0003141*(pdg0001467 - 1)/pdg0001467
8
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 4245712581
    \(v = \frac{2 \pi r}{t}\)
  1. 9346215480
    \(T_{\rm orbit}\)
  1. 3722461713
    \(t\)
  1. 3614055652
    \(v = \frac{2 \pi r}{T_{\rm orbit}}\)
 LHS diff is 0 RHS diff is 2*pdg0002530*pdg0003141*(-pdg0001467 + pdg0008762)/(pdg0001467*pdg0008762)
9
  • 0000111483: raise both sides to power
  • number of inputs: 1; feeds: 1; outputs: 1
  • Raise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.
  1. 3614055652
    \(v = \frac{2 \pi r}{T_{\rm orbit}}\)
  1. 2754264786
    \(2\)
  1. 8059639673
    \(v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
 recognized infrule but not yet supported
10
  • 0000111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 4072200527
    \(\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
  1. 5359471792
    \(\frac{m_{\rm satellite}}{r}\)
  1. 1994296484
    \(v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}\)
 valid
11
  • 0000111355: LHS of expr 1 equals LHS of expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8059639673
    \(v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
  1. 1994296484
    \(v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}\)
  1. 3906710072
    \(G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
 input diff is pdg0001357**2 - pdg0004082**2 diff is 4*pdg0002530**2*pdg0003141**2/pdg0008762**2 - pdg0005458*pdg0006277/pdg0002530 diff is -4*pdg0002530**2*pdg0003141**2/pdg0008762**2 + pdg0005458*pdg0006277/pdg0002530
12
  • 0000111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 3906710072
    \(G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
  1. 6238632840
    \(r T_{\rm orbit}^2\)
  1. 7010294143
    \(T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3\)
 valid
13
  • 0000111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 7010294143
    \(T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3\)
  1. 7556442438
    \(4 \pi^2\)
  1. 4858693811
    \(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3\)
 valid
14
  • 0000111483: raise both sides to power
  • number of inputs: 1; feeds: 1; outputs: 1
  • Raise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.
  1. 4858693811
    \(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3\)
  1. 4319544433
    \(1/3\)
  1. 2617541067
    \(\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r\)
 recognized infrule but not yet supported
15
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 3920616792
    \(T_{\rm geostationary orbit} = 24\ {\rm hours}\)
 no validation is available for declarations
16
  • 0000111984: change two variables in expression
  • number of inputs: 1; feeds: 4; outputs: 1
  • Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.
  1. 2617541067
    \(\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r\)
  1. 7053449926
    \(r_{\rm geostationary\ orbit}\)
  1. 5770088141
    \(r\)
  1. 5208737840
    \(T_{\rm geostationary\ orbit}\)
  1. 3846345263
    \(T_{\rm orbit}\)
  1. 1559688463
    \(\left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit}\)
 LHS diff is 2**(1/3)*(-(pdg0005458*pdg0005595**2*pdg0006277/pdg0003141**2)**(1/3) + (pdg0005458*pdg0006277*pdg0008762**2/pdg0003141**2)**(1/3))/2 RHS diff is pdg0002530 - pdg0007110

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